Engineering Fracture Mechanics 78 (2011) 3270–3288

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Engineering Fracture Mechanics

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Geometric functions of stress intensity factor solutions for spot weldsin cross-tension specimens

P.-C. Lin ⇑, Z.-L. LinAdvanced Institute for Manufacturing with High-tech Innovations & Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi 621, Taiwan

a r t i c l e i n f o

Article history:Received 21 March 2011Received in revised form 20 July 2011Accepted 17 September 2011

Keywords:Spot weldCross-tension specimenStress intensity factorGeometric functionFatigue

0013-7944/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.engfracmech.2011.09.015

⇑ Corresponding author. Tel.: +886 5 272 0411x3E-mail address: imepcl@ccu.edu.tw (P.-C. Lin).

a b s t r a c t

Stress intensity factor solutions for spot welds in cross-tension specimens are investigatedby finite element analyses. Three-dimensional finite element models are developed toobtain accurate solutions. Various ratios of sheet thickness, half specimen width and halfeffective specimen length to nugget radius are considered. The computational results con-firm the functional dependence on the nugget radius and sheet thickness of Zhang’s ana-lytical solutions. The results also provide three geometric functions in terms ofnormalized half specimen width and normalized half effective specimen length to Zhang’sanalytical solutions. Based on the analytical and computational results, the dimensions ofcross-tension specimens and the corresponding approximate stress intensity factor solu-tions are suggested.

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1. Introduction

Resistance spot welding is widely used to join sheet metals for automotive components. The fatigue life of spot welds hasbeen investigated by many researchers in various types of specimens, for example, see Zhang [1]. Since the spot weld pro-vides a natural crack or notch along the nugget circumference, fracture mechanics has been adopted to investigate thefatigue life of spot welds in various types of specimens based on the stress intensity factor solutions at the critical locationsof spot welds. In engineering applications, the stress intensity factor or equivalent stress intensity factor solutions at thecritical locations of spot welds can be directly used to correlate the experimental results of spot welds under cyclic loadingconditions without considerations of the fatigue crack propagation, see Zhang [2,3] and Radaj et al. [4].

Swellam et al. [5] indicated that the fatigue failure process of a spot weld can be divided into three stages. Stage I corre-sponds to the crack initiation and growth when kinked cracks emanate from the critical locations of the main crack (originalcrack) along the nugget circumference up to about 18% of the sheet thickness. Stage II corresponds to the crack propagationthrough the sheet thickness. Stage III corresponds to the crack propagation through the width of specimens. Note that thefatigue life of a spot weld is in general dominated by Stages I and II. Accordingly, the stress intensity factor solutions forkinked cracks are needed to estimate the fatigue life based on the linear elastic fracture mechanics. Bilby et al. [6] and Cotte-rell and Rice [7] indicated that the stress intensity factors of a kink crack with vanishing kink length can be expressed asclosed-form functions of the kink angle and the stress intensity factors of the main crack. Based on the engineering approachof Newman and Dowling [8], the fatigue life of spot welds can be estimated if the stress intensity factors for kinked cracks areassumed to be relatively constant during Stage II. Therefore, the accurate stress intensity factor solutions for the main crackare also essential parameters for the fatigue life estimation of spot welds. In the following, the stress intensity factors for themain crack are denoted as the stress intensity factors for convenience.

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Nomenclature

a nugget radiusCT01,CT02, . . . ,CT04 critical locations with the maximum mode II stress intensity factorCT05,CT06, . . . ,CT08 critical locations with the maximum mode III stress intensity factorE, m Young’s modulus and Poisson’s ratiofII,fIII scaling factors for Lin and Pan’s mode II and mode III stress intensity factor solutionsF opening forceFI, FII, FIII geometric factors for the mode I, mode II and mode III stress intensity factor solutions of cross-tension speci-

mensDF load rangegc, hc geometric parametersK stress intensity factorKI, KII, KIII mode I, mode II and mode III stress intensity factorsDKI, DKII, DKIeq mode I, mode II, and equivalent stress intensity factor rangesL half length of the upper and lower sheets of cross-tension specimensL0 effective half length of the upper and lower sheets of cross-tension specimenseMb bending moment per unit length caused by the longer sheets of the specimeneMc bending moment per unit length caused by the fixture constrainteMx0y0 twisting moment per unit lengthr, h polar coordinatesS fixture lengtht sheet thicknessW half specimen widthW0 equivalent radius for the equivalent circular plate modelx, y, z Cartesian coordinatesX, Y, X0 functions of a, W and mrðÞCross structural stresses near spot welds in cross-tension specimens

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3271

The stress intensity factors of spot welds usually vary point by point along the nugget circumference. In order to correlatethe fatigue life of spot welds, Pook [9,10] gave the maximum stress intensity factor solutions for spot welds in lap-shearspecimens, coach-peel specimens, circular plates and other bending dominant plate and beam specimens. Swellam et al.[11] proposed a stress index Ki by modifying their stress intensity factor solutions to correlate their experimental resultsfor various types of specimens. Zhang [1–3] obtained closed-form stress intensity factor solutions for spot welds in varioustypes of specimens in order to correlate the experimental results of spot welds in these specimens under cyclic loading con-ditions. Recently, Lin et al. [12] developed a new analytical solution for a finite square plate with a rigid inclusion undercounter bending conditions. The new analytical solution successfully modeled the mode I stress intensity factor solutionfor spot welds in lap-shear specimens from the computational results of Wang et al. [13]. Lin and Pan [14] presented a the-oretical framework and closed-form stress intensity factor solutions in terms of structural stresses for spot welds under var-ious types of loading conditions. Lin and Pan [15] then adopted the analytical solutions from Lin and Pan [14] to derive newclosed-form structural stress and stress intensity factor solutions for spot welds in lap-shear, square-cup, U-shape, cross-tension and coach-peel specimens.

In order to obtain accurate stress and strain distributions and/or stress intensity factor solutions for spot welds in variousspecimens, finite element analyses have been carried out by various investigators [13,16–24]. Atzori et al. [16] conductedtwo-dimensional and three-dimensional linear elastic finite element analyses to investigate the effects of spot weld densityon the stress distributions in/near spot welds in lap-shear specimens and correlate the computational results to the fatiguebehavior of spot welds. Satoh et al. [17] conducted three-dimensional elastic and elastic–plastic finite element analyses toinvestigate the stress and strain distributions in the symmetry plane near spot welds in lap-shear specimens to identify thefatigue crack initiation sites under high-cycle and low-cycle fatigue loading conditions. Deng et al. [18] conducted elastic andelastic–plastic three-dimensional finite element analyses to investigate the stress fields in and near the nuggets in lap-shearand symmetrical coach peel specimens to understand the effects of the nugget size and sheet thickness on the interfacial andnugget pull out failure modes. Pan and Sheppard [19] conducted three-dimensional elastic–plastic finite element analyses tocorrelate the fatigue lives of spot welds to the cyclic principal strain ranges of the material element near the main notch inlap-shear and modified coach-peel specimens.

Radaj et al. [20] used simplified finite element models where the plate and brick elements were used for sheets and spotwelds, respectively, to obtain the stress intensity factor solutions along the nugget circumference in various specimens.Zhang [3,21] obtained the stress intensity factor solutions at the selected critical points of spot welds in lap-shear, coach-peel, and cross-tension specimens where the spot welds were treated as rigid and the sheet metals were modeled by plateelements. Pan and Sheppard [22] conducted three-dimensional finite element analyses to investigate the critical local stress

3272 P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288

intensity factor solutions for kinked cracks with a nearly elliptical shape emanating from the main notch along the nuggetcircumference in lap-shear specimens and modified coach-peel specimens. Note that the aforementioned computational re-sults are available for their specimens of specific geometries; however, these results cannot be applied to specimens of dif-ferent designs. Therefore, Wang et al. [13,23] and Lin and Wang [24] systematically investigated the effects of specimengeometry on the stress intensity factor solutions of spot welds in square-cup, lap-shear, and U-shape specimens based onthree-dimensional finite element analyses.

Fig. 1a schematically shows a cross-tension specimen commonly used to investigate the strength and fatigue life of spotwelds under opening-dominated loading conditions. The weld nugget is idealized as a circular cylinder. The applied openingforces F/2 are shown as bold arrows. The overlapped portion of the specimen is marked by dashed lines. The cross-tensionspecimen has the sheet thickness t, the nugget diameter 2a, the specimen width 2W, and the specimen length 2L. The upperand lower sheets have equal thickness. According to the standard of AWS (C1.1M/C1.1:2000) [25], the gray rectangular areasat the ends of the upper and lower sheets should be fully clamped by fixtures to apply opening loads during testing. Thelength of the fixtures is denoted as S. Note that the white rectangular areas of the upper and lower sheets are denoted asthe central effective portion of the specimen since the areas clamped by the fixtures have no contribution to the bendingmoment applied to the spot weld. The length of the central effective portion is therefore denoted as the effective specimenlength 2L0 (=2L � 2S). Fig. 1b schematically shows the top view of a nugget with a cylindrical coordinate system centered atthe nugget center. The critical locations with the maximum mode II stress intensity factor are marked as points CT01, CT02,CT03, and CT04. The critical locations with the maximum mode III stress intensity factor are marked as points CT05, CT06,CT07, and CT08. Note that the mode I stress intensity factor is nearly constant along the nugget circumference and thereforehas no critical location.

In this paper, the stress intensity factor solutions for spot welds in cross-tension specimens are investigated by a system-atic finite element analysis. Three-dimensional finite element models based on the finite element model for two circular

(a)

(b)

Fig. 1. (a) A schematic plot of a cross-tension specimen. The applied opening forces F/2 are shown as bold arrows. The weld nugget is idealized as a circularcylinder. (b) A top view of a nugget with a cylindrical coordinate system centered at the nugget center. The critical locations with the maximum mode IIstress intensity factor are marked as points CT01, CT02, CT03, and CT04. The critical locations with the maximum mode III stress intensity factor are markedas points CT05, CT06, CT07, and CT08.

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3273

plates with a connection [23] are used to obtain the stress intensity factor solutions for cross-tension specimens. The stressintensity factor solutions as functions of the ratios t/a, W/a, and L0/a are investigated. The stress intensity factor solutions ofthese computational results are compared with some existing computational and closed-form analytical stress intensity fac-tor solutions for cross-tension specimens. Geometric functions in terms of the ratios W/a and L0/a based on the computa-tional results are suggested to the stress intensity factor solutions proposed by Zhang [3].

2. Stress intensity factor solutions

Yuuki et al. [26] obtained the stress intensity factor solutions at the critical locations (points CT01, CT02, CT03, and CT04as shown in Fig. 1b) of cross-tension specimens as

KI ¼2:828F

2affiffiffiffiffiffi2ap e½�1:755þ0:936ð2a=tÞ�0:064ð2a=tÞ2 � ð1Þ

KII ¼2:828F

2affiffiffiffiffiffi2ap e½�3:45þ0:827ð2a=tÞ�0:051ð2a=tÞ2 � ð2Þ

Swellam et al. [11] presented that the mode I stress intensity factor solution for spot welds was obtained from the solutionsfor two semi-infinite half spaces connected by a circular patch under an axial force and a moment in Tada et al. [27]. Themode II stress intensity factor solution for spot welds was approximated from the mode III stress intensity factor solutionfor two semi-infinite half spaces connected by a circular patch under a twisting moment in Tada et al. [27]. For cross-tensionspecimens, the stress intensity factor solutions at the critical locations (points CT01, CT02, CT03, and CT04 as shown inFig. 1b) were proposed as

KI ¼2aF þ 3FL0

4a2ffiffiffiffiffiffipap ð3Þ

KII ¼ KIII ¼ 0 ð4Þ

Zhang [3] obtained the stress intensity factor solutions at the critical locations (points CT01, CT02, CT03, and CT04 as shownin Fig. 1b) of cross-tension specimens as

KI ¼3ffiffiffi3p

L0F16pat

ffiffitp ð5Þ

KII ¼3L0F

32patffiffitp ð6Þ

KIII ¼ 0 ð7Þ

Note that these solutions were obtained from the closed-form structural stress solutions for an infinite plate with a rigidinclusion under central bending conditions and J integral formulation. In general, the infinite plate model has a geometryof a square plate. Therefore, the functional forms of Zhang’s K solutions should be valid for cross-tension specimens of thinsheets with finite ratios of W/a and L0/a.

Note that Zhang [3] approximated the counter opening load for the rigid inclusion by a center bending load, namely,a moment applied to the rigid inclusion, to obtain the stress solutions for the derivation of the K solutions. In contrast tothe approximate approach of Zhang [3], Lin and Pan [15] considered another way to decompose the counter openingload for cross-tension specimens. Lin and Pan’s K solutions were obtained from the closed-form stress solutions for arigid inclusion in a finite square plate under opening, cross opening/closing and counter bending conditions, as shownin Fig. 2, and the J integral formulation. Fig. 2 schematically shows the load decomposition for the overlapped portion ofa cross-tension specimen (marked by dashed lines in Fig. 1a). Lin and Pan [15] obtained the stress intensity factor solu-tions at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b) for cross-tension specimens withfinite size as

KI ¼ffiffitpffiffiffi

3p �6W2ð ~Mb � ~McÞ

t2Y� 3F½ða2 �W 02Þð�1þ mÞ þ 2W 02ð1þ mÞ lnðW 0=aÞ�

2pt2½a2ð�1þ mÞ �W 02ð1þ mÞ�

( )ð8Þ

KII ¼ffiffitp

2�12ð ~Mb � ~McÞ

t2Xða4W4 þW8Þ � 24 ~Mx0y0

t2X0½a4ðW=

ffiffiffi2pÞ4 þ ðW=

ffiffiffi2pÞ8�

( )ð9Þ

KIII ¼ 0 ð10Þ

Fig. 2. Decomposition of the load for a cross-tension specimen. Model A represents a spot weld in the overlapped portion of a cross-tension specimen(marked by dashed lines in Fig. 1a) under opening loading conditions. Model B represents the upper half of model A. The forces and moments of model B areapproximately decomposed into three types of loads: counter bending (model C), opening (model D) and cross opening/closing (model E).

3274 P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288

where W0 is the equivalent radius for the equivalent circular plate model and is approximately equal to 2W=ffiffiffiffipp

using thesimple area equivalence rule. Here, X, Y and X0 are defined as

X ¼ ð�1þ mÞða4 þW4Þ2 � 4a2W6ð1þ mÞ ð11aÞY ¼ a2ð�1þ mÞ �W2ð1þ mÞ ð11bÞX0 ¼ ð�1þ mÞ½a4 þ ðW=

ffiffiffi2pÞ4�2 � 4a2ðW=

ffiffiffi2pÞ6ð1þ mÞ ð11cÞ

where m is the Poisson’s ratio of the sheet material. In Eq. (9), eMx0y0 (=F/16) is the uniform distributed twisting moment perunit length. The eMb (=F(L0 �W)/4W) is the additional counter bending moment per unit length due to longer upper and lowersheets of cross-tension specimens. The eMc is the counter bending moment per unit length caused by the constraint of fix-tures and is given as

eMc ¼ gcF½a2 �W 02 þ 2a2 lnðW 0=aÞ�

4pða2 �W 02Þþ hc

~Mb

2ð12Þ

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3275

where gc (=0.5 + 0.5(L0/W)(W/L0)) is a geometric parameter as a function of the specimen width and effective specimen length[24] and hc is another geometric parameter which is applicable when L0 > W. The hc value is defined as 0 for L0 = W and de-fined as 1 for L0 > W. Note that Lin and Pan [15] indicated that the K solutions for cross-tension specimens with finite size areapplicable as L0 P W. The details of the validation of the K solutions in Eqs. (8)–(10), (11a)–(11c), (12) are reported in Lin andPan [15].

Currently, no satisfactory closed-form KIII solution for cross-tension specimens is available in the literature. Lin and Pan[15] indicated that the structural stresses near the spot weld have no contribution to the KIII solution of cross-tension spec-imens since the structural stress solutions are derived from the assumption of rigid inclusions for spot welds. Therefore, Linand Pan’s KIII solution is identically zero along the nugget circumference.

3. Finite element analyses

Obtaining accurate stress intensity factor solutions for spot welds in cross-tension specimens and systematically exam-ining the validity of the stress intensity factor solutions of Yuuki et al. [26], Swellam et al. [11], Zhang [3], and Lin and Pan[15] require three-dimensional finite element computations. Note that finite element analyses for cross-tension specimenswere carried out in Radaj et al. [20] and Zhang [21] for specific ratios of the sheet thickness to the nugget radius, t/a, the halfspecimen width to the nugget radius, W/a, the half effective specimen length to the nugget radius, L0/a. No systematic inves-tigation of the effects of t/a, W/a, and L0/a on the stress intensity factor solutions for cross-tension specimens exists in theliterature. Therefore, the well-benchmarked finite element models developed by Wang et al. [23] are used in this investiga-tion. The finite element models for cross-tension specimens are evolved from the three-dimensional finite element model fortwo circular plates with a connection where a mesh sensitivity study was carried out for benchmark based on Pook’s ana-lytical solution [10] under axisymmetric loading conditions [23]. The details to select an appropriate three-dimensionalmesh for obtaining accurate stress intensity factor solutions for spot welds are reported in Wang et al. [23].

Fig. 3a schematically illustrates a cross-tension specimen subjected to a uniform displacement on the gray areas of theupper sheet (bold arrows) and clamped boundary conditions on the gray areas of the lower sheet. The cylinder representsthe weld nugget. The specimen has a sheet thickness t of 0.65 mm, a half specimen length L of 76.2 mm, a half specimenwidth W of 25.4 mm, and a nugget radius a of 3.2 mm based on the standard of AWS (C1.1M/C1.1:2000) [25]. Both upperand lower sheets have the same thickness. The gray areas at the ends of the specimen require being clamped by fixturesto apply opening loads during testing. The fixtures have a length S of 50.8 mm and a width 2W of 50.8 mm.

Once the specimen is clamped by fixtures, those portions have no contribution to the bending moments applied to thespot weld. Therefore, the cross-tension specimen (Fig. 3a) can be simplified as an equivalent model of two crossed rectan-gular sheets with an effective length of 2L0 joined by a spot weld (Fig. 3b). The equivalent model is subjected to a uniformdisplacement along both ends of the upper sheet (bold arrows), simply supported edge conditions along both ends of thelower sheet, and roller edge conditions along the ends of the upper and lower sheets. The half effective specimen lengthL0 is 25.4 mm. A Cartesian coordinate system is also shown in the figure. As shown in Fig. 3b, the uniform displacement isapplied in the �z direction to the ends of the upper sheet, while the displacement in the x direction for the edges of the uppersheet and that in the y and z directions for the edges of the lower sheet are constrained.

Fig. 3c shows a mesh for a right quarter finite element model. Fig. 3d shows a close-up view of the mesh near the maincrack tip. Note that the main crack is modeled as a sharp crack here. The three-dimensional finite element mesh near theweld nugget is evolved from that for two circular plates with a connection, as discussed in Wang et al. [23]. As shown inFig. 3d, the mesh near the center of the weld nugget is refined to ensure reasonable aspect ratios of the three-dimensionalbrick elements. The three-dimensional finite element model for a cross-tension specimen has 46,536 20-node quadraticbrick elements. The main crack surfaces are shown as bold lines in Fig. 3d.

In this work, the weld nugget and the base metal are assumed to be linear elastic isotropic materials. The Young’s mod-ulus E is taken as 200 GPa and the Poisson’s ratio m is taken as 0.3. The commercial finite element program ABAQUS [28] isemployed to perform the computations. Brick elements with quarter point nodes and collapsed nodes along the crack frontare adopted to model the 1=

ffiffiffirp

singularity near the crack tip.The distributions of the stress intensity factor solutions along the nugget circumference of cross-tension specimens are

first investigated. Fig. 4 shows the normalized KI, KII, and KIII solutions as functions of h for the crack front along the nuggetcircumference based on the three-dimensional finite element computational results for a cross-tension specimen witht/a = 0.2, W/a = 7.94, and L0/a = 7.94. The orientation angle h is measured clockwise from the critical location of pointCT01, as shown in Fig. 1b. The computational results, denoted as KI;KII , and KIII , are normalized by the average value ofthe KI solution. Note that the KI solution is nearly constant along the crack front, whereas the KII, and KIII solutions vary pointby point along the crack front. As shown in Fig. 4, the minima of the KII solution are located at points CT01 (h = 0�) and CT03(h = 180�) and the maxima of the KII solution are located at points CT02 (h = 90�) and CT04 (h = 270�). The maxima of the KIII

solution are located at points CT05 (h = 45�) and CT07 (h = 225�) and the minima of the KIII solution are located at points CT06(h = 135�) and CT08 (h = 315�). As shown in Fig. 4, the average value of the KI solution is nearly 5.6 and 7.18 times higher thanthe maximum values of the KII and KIII solutions, respectively. The computational results indicate that the stress intensityfactor KI is the dominant stress intensity factor of cross-tension specimens under opening loading conditions. Note thatthe distributions of the stress intensity factor solutions along the nugget circumference are similar to those in Radaj et al.[20].

(a) (b)

(d)

(c)

Fig. 3. (a) A cross-tension specimen subjected to a uniform displacement on the gray areas of the upper sheet (bold arrows) and clamped boundaryconditions on the gray areas of the lower sheet. The cylinder represents the weld nugget. (b) An equivalent model subjected to a uniform displacementalong both ends of the upper sheet (bold arrows), simply supported edge conditions along both ends of the lower sheet, and roller edge conditions along theends of the upper and lower sheets. (c) A mesh for a right quarter finite element model. (d) A close-up view of the mesh near the main crack tip.

3276 P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288

As shown in Fig. 4, the KI solution is nearly constant along the nugget circumference. Zhang [2,3] indicated that the KI

solution depends on the symmetric structural stress rSymmCross with respect to the interfacial surface of the nugget, while the

KII solution depends on the anti-symmetric structural stress rAntiCross. Lin and Pan [15] indicated that for cross-tension speci-

mens, the structural stress near the spot weld in the upper sheet rUpperCross is a function of cos 2h, while that in the lower sheet

rLowerCross is a function of cos (2h + p). The rSymm

Cross solution, which correlates with the KI solution, can be obtained byðrUpper

Cross þ rLowerCross Þ=2. All terms containing cos 2h are canceled out. Therefore, Lin and Pan’s KI solution for cross-tension spec-

imens is theoretically uniform along the nugget circumference. However, due to the finite ratios of W/a and L0/a, the com-putational KI solution is nearly constant along the nugget circumference. On the other hand, the anti-symmetric structuralstress rAnti

Cross, which correlates with the KII solution, can be obtained by ðrUpperCross � rLower

Cross Þ=2. The KII solution for cross-tensionspecimens is a function of h.

W/a=7.94L'/a=7.94

t/a=0.2 KI

KII

KIII

θ (degree)

0 45 90 180135 225 270 315 360

KI/K

I, K

II/K

I, K

III/

KI

1

0

1.4

1.2

0.8

0.6

0.4

0.2

-0.2

-0.4

Fig. 4. The normalized KI, KII, and KIII solutions as functions of h for the crack front along the nugget circumference based on the three-dimensional finiteelement computational results for a cross-tension specimen with t/a = 0.2, W/a = 7.94, and L0/a = 7.94.

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3277

3.1. Normalized sheet thickness t/a

The effects of t/a on the KI solution, the KII solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown inFig. 1b), and the KIII solution at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) for the spec-imens with W/a = 7.94 and L0/a = 7.94 are investigated. Fig. 4 shows that the KI solution is nearly constant along the nuggetcircumference. The maximum magnitudes of the KII solution occur at points CT01, CT02, CT03, and CT04. The maximummagnitudes of the KIII solution occur at points CT05, CT06, CT07, and CT08. Different three-dimensional finite element mod-els are developed by changing the sheet thickness t with the nugget radius a being fixed at 3.2 mm.

Fig. 5 shows the normalized KI solutions as functions of t/a based on the three-dimensional finite element computationsand the analytical solutions of Yuuki et al. [26] in Eq. (1), Swellam et al. [11] in Eq. (3), Zhang [3] in Eq. (5), and Lin and Pan[15] in Eq. (8) under the applied load F. The KI solutions are normalized by Zhang’s KI solution in Eq. (5), denoted as (KI)Zhang.The normalized KI solutions based on the computational results and the analytical solutions of Yuuki et al. [26] in Eq. (1),

W/a=7.94L'/a=7.94

1.4

1.2

0.8

0.6

0.4

0.2

1

0

1.6

1.8

2

t/a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KI _Zhang~KI _FEM~

KI _Lin&Pan~

KI _Yuuki~

KI _Swellam~

KI/(

KI)

Zha

ng

Fig. 5. The normalized KI solutions as functions of t/a based on the three-dimensional finite element computations and the analytical KI solutions for cross-tension specimens. The normalized KI solutions based on the computational results, the analytical solutions of Yuuki et al. [26] in Eq. (1), Swellam et al. [11]in Eq. (3), Zhang [3] in Eq. (5), and Lin and Pan [15] in Eq. (8) are denoted as eK I FEM; eK I Yuuki; eK I Swellam; eK I Zhang, and eK I Lin&Pan, respectively.

3278 P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288

Swellam et al. [11] in Eq. (3), Zhang [3] in Eq. (5), and Lin and Pan [15] in Eq. (8) are denoted aseK I FEM; eK I Yuuki; eK I Swellam; eK I Zhang, and eK I Lin&Pan, respectively. The range of the ratio t/a is selected from 0.08 to0.60 to represent the typical values for spot welds used in the automotive industry. Note that the values of gc and hc in

Eq. (12) for eK I Lin&Pan are 1.0 and 0, respectively, for the specimens here. Fig. 5 shows that eK I FEM based on the three-dimensional finite element computations appears to be relatively constant for the ratios of t/a considered. The general trends

of eK I Zhang and eK I Lin&Pan agree well with that of eK I FEM. The average of eK I FEM is nearly 15% higher than eK I Zhang but

18% lower than eK I Lin&Pan. However, the general trends of eK I Yuuki and eK I Swellam are quite different from that of eK I FEM.eK I Yuuki shows a significant deviation from eK I FEM as t/a decreases from 0.3. eK I Swellam also shows a significant deviation

from eK I FEM for the ratios of t/a considered.Fig. 5 shows that eK I FEM confirms the functional dependence of eK I Zhang and eK I Lin&Pan on the sheet thickness t and the

nugget radius a. Both the analytical KI solutions of Zhang [3] and Lin and Pan [15] can be used as the basis to express the KI

solution for cross-tension specimens. However, the KI solution of Lin and Pan [15] is rather complex and apparently cannotbe easily used for engineering applications. Therefore, the KI solution of Zhang [3] will be used later. Note that the KI solutionof Zhang [3] is obtained from the closed-form stress solutions for a rigid inclusion under center bending conditions in aninfinite plate and the J integral formulation. For a cross-tension specimen with a finite width 2W and a finite effective length2L0, a deviation from the analytical KI solution of Zhang [3] is possible for finite ratios of W/a and L0/a. Therefore, the KI solu-tion for a cross-tension specimen with finite ratios of W/a and L0/a can be written as

Fig. 6.dimenscomputeK II FEM

ðKIÞcross-tension ¼ FIðW=a; L0=aÞ 3ffiffiffi3p

L0F16pat

ffiffitp ð13Þ

where FI is a geometric factor as a function of W/a and L0/a. For a specimen with t/a = 0.20, W/a = 7.94, and L0/a = 7.94 basedon the standard of AWS (C1.1M/C1.1:2000) [25], the FI value is 1.18. Therefore, the geometric factor FI is needed.

Fig. 6 shows the normalized KII solutions at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b)as functions of t/a based on the three-dimensional finite element computations and the analytical solutions of Yuuki et al.[26] in Eq. (2), Zhang [3] in Eq. (6), and Lin and Pan [15] in Eq. (9) under the applied load F. The KII solutions are normalized byZhang’s KII solution in Eq. (6), denoted as (KII)Zhang. The normalized KII solutions based on the computational results and theanalytical solutions of Yuuki et al. [26] in Eq. (2), Zhang [3] in Eq. (6), and Lin and Pan [15] in Eq. (9) are denoted aseK II FEM; eK II Yuuki; eK II Zhang, and eK II Lin&Pan, respectively. Note that the values of gc and hc in Eq. (12) for eK II Lin&Pan are1.0 and 0, respectively. Fig. 6 shows that eK II FEM slightly decreases as t/a increases. The general trends of eK II Zhang andeK II Lin&Pan agree with that of eK II FEM. The average of eK II FEM is nearly 41% and 73% lower than eK II Zhang andeK II Lin&Pan, respectively. On the other hand, the general trend of eK II Yuuki is quite different from that of eK II FEM. As t/a de-creases from 0.3, eK II Yuuki shows a significant deviation from eK II FEM. Note that the KII solution of Swellam et al. [11] in Eq.(4) is identically zero and not considered here.

W/a=7.94L'/a=7.94

1.4

1.2

0.8

0.6

0.4

0.2

1

0

1.6

1.8

2

KII_Zhang~KII_FEM~

KII_Lin&Pan~

KII_Yuuki~

t/a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

KII

/(K

II) Z

hang

The normalized KII solutions at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b) as functions of t/a based on the three-ional finite element computations and the analytical KII solutions for cross-tension specimens. The normalized KII solutions based on theational results, the analytical solutions of Yuuki et al. [26] in Eq. (2), Zhang [3] in Eq. (6), and Lin and Pan [15] in Eq. (9) are denoted as; eK II Yuuki; eK II Zhang, and eK II Lin&Pan, respectively.

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3279

Although eK II FEM shows a slight dependence on t/a, eK II FEM still confirms the functional dependence of eK II Zhang andeK II Lin&Pan on the sheet thickness t and the nugget radius a. Consequently, both the analytical KII solutions of Zhang [3]and Lin and Pan [15] can be used as the basis to express the KII solution of cross-tension specimens. However, the deviationof Lin and Pan’s KII solution from the computational results is relatively high. Therefore, Zhang’s KII solution will be laterused. As discussed earlier, a deviation from Zhang’s KII solution is still possible for finite ratios of W/a and L0/a. Accordingly,the KII solution for a cross-tension specimen with finite ratios of W/a and L0/a can be written as

Fig. 7.dimens

ðKIIÞcross-tension ¼ FIIðW=a; L0=aÞ 3L0F32pat

ffiffitp ð14Þ

where FII is a geometric factor as a function of W/a and L /a. Here, the FII value is 0.73 for a specimen with t/a = 0.20,W/a = 7.94, and L0/a = 7.94. Therefore, the geometric factor FII is needed.

Fig. 7 shows the normalized KIII solution at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) asa function of t/a based on the three-dimensional finite element computations under the applied load F. Note that the com-putational KIII solution is normalized by Zhang’s KII solution in Eq. (6), denoted as (KII)Zhang. Zhang’s KII solution is consideredhere since no closed-form KIII solution for cross-tension specimens is available in the literature. In addition, the generaltrends and magnitudes of the KII and KIII solutions along the nugget circumference are similar, except a phase differenceof p/4 exists between them, as shown in Fig. 4. Therefore, (KII)Zhang is here regarded as a substitution for the KIII solution.As shown in Fig. 7, the normalized KIII solution, eK III FEM, based on the three-dimensional finite element computations ap-pears to be relatively constant for the ratios of t/a considered. The average of eK III FEM is 0.57.

Although no closed-form KIII solution is available, eK III FEM still confirms the functional dependence of Zhang’s KII solutionon the sheet thickness t and the nugget radius a. Therefore, Zhang’s KII solution will be later used as the basis to express theKIII solution. Similarly, a deviation from Zhang’s KII solution to the computational KIII solutions is quite possible for finite ra-tios of W/a and L0/a. Accordingly, the KIII solution for a cross-tension specimen with finite ratios of W/a and L0/a can be writtenas

ðKIIIÞcross-tension ¼ FIIIðW=a; L0=aÞ 3L0F32pat

ffiffitp ð15Þ

where FIII is a geometric factor as a function of W/a and L0/a. Note that FIII can be considered as a factor which transformsZhang’s KII solution into a new KIII solution. Here, the FIII value is 0.57 for a specimen with t/a = 0.20, W/a = 7.94, andL0/a = 7.94. Therefore, the geometric factor FIII is needed.

As shown in Figs. 5–7, eK I FEM and eK III FEM show nearly no dependence on t/a, while eK II FEM shows a slight dependenceon t/a. This suggests that, as t/a increases, the thickness effects increase and the computational solutions slightly deviatefrom those analytical solutions based on the thin plate theory. Validating the closed-form K solutions based on the thin platetheory requires more attention being paid on the computational solutions for small t/a ’s. The investigation is continuedbased on the assumption that the geometry factors FI, FII, and FIII have a weak dependence on t/a for a small t/a value of0.20. As can be seen in Eqs. (13)–(15), FI, FII, and FIII are functions of the two normalized geometric parameters W/a andL0/a. The number of computations is too large for a full-scale numerical investigation of FI, FII, and FIII on the complex coupling

W/a=7.94L'/a=7.94

KII

I/(K

II) Z

hang

1.4

1.2

0.8

0.6

0.4

0.2

1

0

1.6

1.8

2KIII_FEM~

t/a0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

The normalized KIII solution at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) as a function of t/a based on the three-ional finite element computations, denoted as eK III FEM.

3280 P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288

effects of W/a and L0/a. In the next section, in order to understand the effects of W/a on the geometric factors FI, FII, and FIII

without significant coupling effects from L0/a, a small L0/a value of 7.94 is chosen. The main reason is that the effects of thecounter bending moment caused by the effective specimen length on these factors are supposedly minimized for small val-ues of L0/a. Therefore, finite element models in the next section are developed for specimens with the ratios of t/a = 0.20 andL0/a = 7.94.

3.2. Normalized half specimen width W/a

The effects of W/a on the geometric factors FI, FII, and FIII are then investigated. Fig. 8 shows the normalized KI solution, thenormalized KII solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b), and the normalizedKIII solution at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) as functions of W/a based on thethree-dimensional finite element computations for t/a = 0.20 and L0/a = 7.94. Note that the KI solution is nearly constantalong the nugget circumference. The normalized KI, KII, and KIII solutions based on the computational results are denotedas eK I FEM; eK II FEM, and eK III FEM, respectively. The KI solution is normalized by Zhang’s KI solution in Eq. (5), whereas theKII and KIII solutions are normalized by Zhang’s KII solution in Eq. (6). Therefore, eK I FEM; eK II FEM, and eK III FEM representthe geometric factors FI, FII, and FIII for t/a = 0.20 and L0/a = 7.94 according to Eqs. (13)–(15), respectively. Note that a smallvalue of t/a = 0.20 is selected to avoid the effects of the thickness. The range of the ratio W/a is selected from 2.0 to 16.0. Notethat some ratios of W/a are larger than the ratio of L0/a (=7.94). Although the models with these ratios may not be applicablein practice, their results can provide a complete understanding to the effects of W/a. Different three-dimensional finite ele-ment models are developed by changing the half specimen width W with the nugget radius a being fixed at 3.2 mm.

Fig. 8 shows that FI decreases slightly when W/a increases and then levels off when W/a is greater than 4.0. On the otherhand, the general trends of FII and FIII are quite similar for the ratios of W/a considered. As W/a increases from 2.0, FII and FIII

decrease significantly. As 6.0 < W/a < 10.0, the rate of decrease slows down as W/a increases. As W/a increases from 10.0, FII

and FIII become stable when W/a increases further. Additionally, the differences between FI, FII, and FIII are nearly constantwhen W/a is greater than or equal to 10.0. As 10.0 < W/a < 16.0, FI, FII, and FIII are nearly 1.18, 0.58, and 0.46, respectively,which indicate the differences from the analytical solutions of Zhang [3].

Fig. 9 shows the normalized KII and KIII solutions as functions of W/a based on our finite element computations, denoted asKII FEM and KIII FEM, respectively. The KII and KIII solutions are normalized by the corresponding KI solutions for the corre-sponding W/a’s. As shown in Fig. 9, KII FEM and KIII FEM are relatively constant when W/a is greater than 10.0. When W/adecreases from 10.0, KII FEM and KIII FEM increase significantly. As W=a ¼ 2:0;KII FEM and KIII FEM are 0.98 and 0.80, respec-tively. The results indicate that the KII and KIII solutions tend to be quite close to the corresponding KI solutions for smallW/a’s. Therefore, the mixture of the modes at the critical locations apparently change significantly when the spacing of spotwelds decreases. Care should be taken for using the KI, KII, and KIII solutions of closely spaced spot welds.

As shown in Fig. 8, the normalized KI, KII, and KIII solutions do not change significantly when W/a is greater than or equalto 10.0. In the next section, in order to understand the effects of L0/a on the geometric factors FI, FII, and FIII without significantcoupling effects from W/a, a large W/a value of 10.0 is chosen since the effects of the specimen width on these factors are

t/a=0.2L'/a=7.94

FI,

FII

, FII

I

W/a

KII_FEM~KI_FEM~

KIII_FEM~

0

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18

Fig. 8. The normalized KI solution, the normalized KII solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b), and thenormalized KIII solution at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) as functions of W/a based on the three-dimensionalfinite element computations for t/a = 0.20 and L0/a = 7.94. The KI, KII, and KIII solutions are normalized by the analytical solutions of Zhang [3], denoted aseK I FEM; eK II FEM, and eK III FEM, respectively.

0

0.2

0.4

0.6

0.8

1

1.2t/a=0.2L'/a=7.94

KII/K

I, K

III/K

I

KII _FEM

KIII_FEM

W/a0 2 4 6 8 10 12 14 16 18

Fig. 9. The normalized KII solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b) and the normalized KIII solution at thecritical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) as functions of W/a based on our finite element computations for t/a = 0.20 andL0/a = 7.94. The KII and KIII solutions are normalized by the corresponding KI solutions, denoted as KII FEM and KIII FEM, respectively.

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3281

supposedly minimized. Therefore, finite element models in the next section are developed for specimens with the ratios oft/a = 0.20 and W/a = 10.0.

3.3. Normalized half effective specimen length L0/a

The effects of L0/a on the geometric factors FI, FII, and FIII are finally investigated. The three-dimensional finite elementmodels have different values of the half effective specimen length L0 with the half specimen width W, nugget radius a,and sheet thickness t being fixed. Fig. 10 shows the normalized KI solution, the normalized KII solution at the critical locations(points CT01, CT02, CT03, and CT04 as shown in Fig. 1b), and the normalized KIII solutions at the critical locations (pointsCT05, CT06, CT07, and CT08 as shown in Fig. 1b) as functions of L0/a based on the three-dimensional finite element compu-tations for t/a = 0.20 and W/a = 10.0. The range of L0/a is selected from 3.0 to 210.0 for Fig. 10a and from 3.0 to 30.0 forFig. 10b. The corresponding ratios of L0/W are also listed below for convenience. Note that some ratios of L0/a are smaller thanthe ratio of W/a (=10.0). These models may not be applicable in practice but their results can provide a complete understand-ing to the effects of L0/a. The normalized KI, KII, and KIII solutions based on the computational results are denoted aseK I FEM; eK II FEM, and eK III FEM, respectively. The KI solution is normalized by Zhang’s KI solution in Eq. (5), whereas the KII

and KIII solutions are normalized by Zhang’s KII solution in Eq. (6). Therefore, the normalized KI, KII, and KIII solutions hererepresent the geometric factors FI, FII, and FIII for t/a = 0.20 and W/a = 10.0 according to Eqs. (13)–(15), respectively.

Fig. 10a shows that FI decreases significantly when L0/a increases from 3.0. As 30.0 < L0/a < 70.0, the rate of decrease slowsdown as L0/a increases. When L0/a increases from 70.0, FI tends to decrease slightly as L0/a increases further. On the otherhand, the general trends of FII and FIII are similar for the ratios of L0/a considered. When L0/a increases from 3.0, FII and FIII

rapidly decrease to the minimum values at L0/a = 10 (L0/W = 1.0) and then gradually increase. The general trends of FII andFIII for small ratios of L0/a (3.0 < L0/a < 30.0) can be seen clearly in Fig. 10b. As 30.0 < L0/a < 110.0, the rate of increase slowsdown as L0/a increases. When L0/a increases from 110.0, FII and FIII increase slightly as L0/a increases further. Note that FI tendsto decrease slightly while FII and FIII tend to increase slightly as L0/a increases further. This indicates that the effects of spec-imen length or fixture constraint on FI, FII, and FIII tend to fade away for large L0/a’s.

As shown in Fig. 10, the computational results deviate significantly from the analytical solutions of Zhang [3] for largeL0/a’s. Note that Zhang [3] used the analytical stress solution for a rigid inclusion in an infinite plate under center bendingconditions to approximate that under counter bending conditions for the KI and KII solutions of cross-tension specimens.The overestimation of the KI solution and the underestimation of the KII solution for large ratios of L0/a’s based on Zhang’sanalytical solutions can be attributed to this approximation. In contrast to the approximate approach of Zhang [3], Linand Pan [15] obtained the closed-form stress solutions from a rigid inclusion in a finite square plate under opening, crossopening/closing and counter bending conditions, as shown in Fig. 2, and the J integral formulation.

Fig. 11a shows the normalized KI solution, the normalized KII solution at the critical locations (points CT01, CT02, CT03,and CT04 as shown in Fig. 1b), and the normalized KIII solutions at the critical locations (points CT05, CT06, CT07, and CT08 asshown in Fig. 1b) as functions of L0/a based on the three-dimensional finite element computations for t/a = 0.20 andW/a = 10.0, and the analytical solutions of Lin and Pan [15]. The normalized KI, KII, and KIII solutions based on the

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2t/a=0.2W/a=10.0

L'/a (L'/W)

KII_FEM~KI _FEM~

KIII_FEM~

0(0)

20(2)

40(4)

60(6)

80(8)

100(10)

120(12)

140(14)

180(18)

160(16)

200(20)

220(22)

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0(0)

5(0.5)

10(1)

15(1.5)

20(2)

25(2.5)

30(3)

35(3.5)

t/a=0.2W/a=10.0

L'/a (L'/W)

KII _FEM~KI _FEM~

KIII_FEM~

(b)

FI,

FII

, FII

IF

I, F

II, F

III

Fig. 10. The normalized KI solution, the normalized KII solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b), and thenormalized KIII solution at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) as functions of L0/a based on the three-dimensionalfinite element computations for t/a = 0.20 and W/a = 10.0. The KI, KII, and KIII solutions are normalized by the analytical solutions of Zhang [3], denoted aseK I FEM; eK II FEM, and eK III FEM, respectively. The ranges of L0/a are (a) 3.0 < L0/a < 210.0 and (b) 3.0 < L0/a < 30.0.

3282 P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288

computational results are denoted as eK I FEM; eK II FEM, and eK III FEM, respectively. The normalized KI and KII solutions basedon the analytical solutions of Lin and Pan [15] in Eqs. (8) and (9) are denoted as eK I Lin&Pan and eK II Lin&Pan, respectively.Note that the analytical solutions of Lin and Pan [15] are only available for L0/W P 1.0. The KI solutions are normalized byZhang’s KI solution in Eq. (5), whereas the KII and KIII solutions are normalized by Zhang’s KII solution in Eq. (6). Therefore,the normalized KI, KII, and KIII solutions here represent the geometric factors FI, FII, and FIII for t/a = 0.20 and W/a = 10.0according to Eqs. (13)–(15), respectively. Fig. 11a shows that the general trend of eK I Lin&Pan agrees well with that ofeK I FEM. As L0/a P 10.0 or L0/W P 1.0, the KI solutions of the finite element computations and Lin and Pan [15] are very closeto each other. Interestingly, the general trend of eK II Lin&Pan also agrees well with that of eK II FEM and even with that ofeK III FEM. However, eK II Lin&Pan shows significant deviations from both eK II FEM and eK III FEM. Based on the curve fitting ap-proach, eK II FEM and eK III FEM can be estimated by eK II Lin&Pan with two scaling factors fII and fIII of 0.57 and 0.43, respectively.As shown in Fig. 11b, the modified KII and KIII solutions of Lin and Pan [15] agree well with eK II FEM and eK III FEM, respectively,

t/a=0.2W/a=10.0

L'/a (L'/W)

KII_FEM~KI_FEM~

KIII_FEM~

0(0)

20(2)

40(4)

60(6)

80(8)

100(10)

120(12)

140(14)

180(18)

160(16)

200(20)

220(22)

F I, F

II, F

III

KII_Lin&Pan~KI_Lin&Pan~

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

L'/a>10.0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2t/a=0.2W/a=10.0

L'/a (L'/W)

KII_FEM~KI_FEM~

KIII_FEM~

0(0)

20(2)

40(4)

60(6)

80(8)

100(10)

120(12)

140(14)

180(18)

160(16)

200(20)

220(22)

F I, F

II, F

III

KII_Lin&Pan*fII~KI_Lin&Pan~

KII_Lin&Pan*fIII~

L'/a>10.0

(b)

(a)

Fig. 11. The normalized KI solution, the normalized KII solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b), and thenormalized KIII solutions at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) as functions of L0/a based on the three-dimensionalfinite element computations for t/a = 0.20 and W/a = 10.0. The computational results are compared with (a) the analytical solutions of Lin and Pan [15] inEqs. (8) and (9), and (b) the modified solutions of Lin and Pan [15] in Eqs. (16)–(18).

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3283

as L0/a P 10.0 or L0/W P 1.0. The effects of L0/a appear to be considered well by the analytical solutions of Lin and Pan [15].The analytical solutions of Lin and Pan [15] with appropriate scaling factors fII and fIII are suggested as the approximateclosed-form solutions for cross tension specimens at the critical locations as.

KI ¼ffiffitpffiffiffi

3p �6W2ð ~Mb � ~McÞ

t2Y� 3F½ða2 �W 02Þð�1þ mÞ þ 2W 02ð1þ mÞ lnðW 0=aÞ�

2pt2½a2ð�1þ mÞ �W 02ð1þ mÞ�

( )ð16Þ

KII ¼fII

ffiffitp

2�12ð ~Mb � ~McÞ

t2Xða4W4 þW8Þ � 24 ~Mx0y0

t2X0½a4ðW=

ffiffiffi2pÞ4 þ ðW=

ffiffiffi2pÞ8�

( )ð17Þ

KIII ¼fIII

ffiffitp

2�12ð ~Mb � ~McÞ

t2Xða4W4 þW8Þ � 24 ~Mx0y0

t2X 0½a4ðW=

ffiffiffi2pÞ4 þ ðW=

ffiffiffi2pÞ8�

( )ð18Þ

KII

/KI,

KII

I/K

I

KII _FEM

KIII_FEM

t/a=0.2W/a=10.0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

L'/a (L'/W)

0(0)

20(2)

40(4)

60(6)

80(8)

100(10)

120(12)

140(14)

180(18)

160(16)

200(20)

220(22)

(a)

KII_FEM

KIII_FEM

t/a=0.2W/a=10.0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0(0)

5(0.5)

10(1)

15(1.5)

20(2)

25(2.5)

30(3)

35(3.5)

L'/a (L'/W)

(b)

KII

/KI,

KII

I/K

I

Fig. 12. The normalized KII solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b) and the normalized KIII solution at thecritical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) as functions of L0/a based on our finite element computations for t/a = 0.20 and W/a = 10.0. The KII and KIII solutions are normalized by the corresponding KI solutions, denoted as KII FEM and KIII FEM, respectively. The ranges of L0/a are (a)3.0 < L0/a < 210.0 and (b) 3.0 < L0/a < 30.0.

3284 P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288

Fig. 12 shows the normalized KII and KIII solutions as functions of L0/a based on our finite element computations, denoted asKII FEM and KIII FEM, respectively. The KII and KIII solutions are normalized by the corresponding KI solutions for the corre-sponding L0/a ’s. The corresponding ratios of L0/W are also listed below for convenience. Fig. 12a shows the computationalresults for 3.0 < L0/a < 210.0 while Fig. 12b shows those for 3.0 < L0/a < 30.0. As shown in Fig. 12a, the general trends ofKII FEM and KIII FEM are quite similar. As L0/a increases from 3.0, KII FEM and KIII FEM decrease slightly to the minimum val-ues at L0/a = 8.0 (L0/W = 0.8) and then gradually increase, as shown in Fig. 12b. This suggests that the spot weld is subjected toopening-dominated loading conditions when the white rectangular areas of the sheets, as shown in Fig. 1a, are nearly square.As L0/a further increases, KII FEM and KIII FEM approach 1.0 at L0/a = 60 and 110, respectively, as shown in Fig. 12a. Note thatwhen the effective specimen length increases, the counter bending moment eMb (=F(L0 �W)/4W) caused by the longer sheetsof the specimen increases. Consequently, the normalized K_II and K_III increase due to the increase of the counter bendingmoment. The mixture of the modes changes significantly for large L0/a’s. Care should be taken for using the KI, KII, and KIII

solutions of long or large cross-tension specimens.

P.-C. Lin, Z.-L. Lin / Engineering Fracture Mechanics 78 (2011) 3270–3288 3285

4. Discussions

Based on the results of finite element computations, the functional dependence of the geometric factors FI, FII, and FIII onW/a and L0/a is investigated under the conditions with minimum coupling effects of W/a and L0/a. The influence of W/a’s onthe geometric factors FI, FII, and FIII is weak for W/a > 10.0 (Fig. 8). The influence of L0/a’s on the geometric factors FI, FII, and FIII

is weak for L0/a > 110.0 (Fig. 10a). Consequently, for a cross-tension specimen with W/a > 10.0 and L0/a > 110.0, the depen-dence of the geometric factors FI, FII, and FIII on W/a and L0/a is minimum. However, the mixture of the modes at the criticallocations of spot welds changes significantly for large L0/a’s. Note that the cross-tension specimen is mainly used to charac-terize the mechanical properties of spot welds subjected to opening loads. Figs. 10b and 12b show that the spot weld is underopening-dominated loading conditions only when the white rectangular areas of the sheets (Fig. 1a) are nearly square.Therefore, a cross-tension specimen with W/a > 10.0 and L0/W = 1.0 is proposed for engineering applications. This can be usedas a design guideline for cross-tension specimens in the future.

A comparison of the existing computational stress intensity factor solutions with the computational results mentioned inthe previous section is then considered. The results from simplified finite element computations of Radaj et al. [20] and Zhang[21] are selected. Table 1 shows that the KI, KII, and KIII solutions at the critical locations obtained from the computation resultsshown in Fig. 4 (listed as Lin & Lin), Radaj et al. [20], and Zhang [21] are normalized by the analytical solutions of Zhang [21]without consideration of the effects of t/a, W/a, and L0/a. Therefore, some normalized KI and KII solutions show significant dif-ferences between each other. The validity of the three solutions cannot be judged easily here. The effects of W/a and L0/ashould be considered. Note that these normalized KI, KII, and KIII solutions are also quite different from 1.0, which implies thatthe deviations between the analytical solutions of Zhang [3] and their computational results are quite significant.

Table 2 lists the FI, FII, and FIII values for t/a, W/a, and L0/a of the specimens used in Fig. 4, Radaj et al. [20], and Zhang [21].Note that the aspect ratio L0/W of these specimens are all identical to 1.0. Based on the interpolation approach proposed inWang et al. [13], the geometry factors FI, FII, and FIII are interpolated by a two-step approach to consider the effects of W/a andL0/a. Note that the interpolation approach is approximate in nature since no result can be used to account for the couplingeffects of W/a and L0/a in this work. First, the initial FI, FII, and FIII values for a given W/a are obtained from Fig. 8 by linearinterpolation to account for the effects of W/a. Note that the results in Fig. 8 are for L0/a = 7.94 and t/a = 0.2. Then, the initial FI,FII, and FIII values are multiplied by the ratios of FI, FII, and FIII for a given L0/a to FI, FII, and FIII for L0/a = 7.94, respectively. Theabove FI, FII, and FIII values are interpolated from the results in Fig. 10 for W/a = 10.0 and t/a = 0.2. Table 2 lists the FI, FII, andFIII values for the three cases. The FI values are moderately different from 1.0 whereas the FII, and FIII values are much smallerthan 1.0 for the three cases. The FI, FII, and FIII values for the specimens used in Fig. 4 and Radaj et al. [20] are nearly identicalbut different from those for the specimen of Zhang [21]. The main reason is that the specimen of Zhang [21] has differentratios of W/a and L0/a.

Table 1The normalized KI, KII, and KIII solutions at the critical locations by the analytical solutions of Zhang [3]from the computational results in Fig. 4 (listed as Lin and Lin), Radaj et al. [20], and Zhang [21].

KI/(KI)Zhang KII/(KII)Zhang KIII/(KII)Zhang

Lin and Lin 1.18 0.73 0.57Radaj et al. [20] 1.17 0.83 N/AZhang [21] 0.90 0.80 N/A

Table 2The FI, FII, and FIII values for different specimen geometries in Fig. 4, Radaj et al. [20], and Zhang [21].

t/a W/a L0/a FI FII FIII

Lin and Lin 0.20 7.94 7.94 1.18 0.73 0.57Radaj et al. [20] 0.40 8.00 8.00 1.17 0.73 0.57Zhang [21] 0.57 14.29 14.29 0.88 0.70 0.59

Table 3The normalized KI, KII, and KIII solutions at the critical locations by FI(KI)Zhang, FII(KII)Zhang, andFIII(KII)Zhang, respectively, for different specimen geometries in Fig. 4, Radaj et al. [20], and Zhang[21].

KI/FI(KI)Zhang KII/FII(KII)Zhang KIII/FIII(KII)Zhang

Lin and Lin 1.00 1.00 1.00Radaj et al. [20] 0.99 1.13 N/AZhang [21] 1.02 1.14 N/A

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Table 3 lists the KI, KII, and KIII solutions at the critical locations normalized by FI(KI)Zhang, FII(KII)Zhang, and FIII(KII)Zhang,respectively. The normalized K solutions of Lin and Lin are all identical to 1.0, which indicates that the K solutions obtaineddirectly from the finite element model in Fig. 4 agree well with those obtained from different finite element models. Thisself-examination also indicates that the geometric factors and the interpolation approach work well. As listed in Table 3,the normalized KI solutions of Radaj et al. [20] and Zhang [21] are nearly identical to 1.0, which indicates that their KI solu-tions based on the simplified finite element computations also agree well with our computational results based on the three-dimensional finite element analyses. This also indicates that the geometric factors FI based on the computational results areindeed needed and useful to scale the KI solutions.

As listed in Table 3, the normalized KII solutions of Radaj et al. [20] and Zhang [21] are 1.13 and 1.14, respectively. Thisindicates that their KII solutions based on simplified finite element computations appear to have 13 and 14 percent overes-timations, respectively. The computational results from Radaj et al. [20] and Zhang [21] appear to have similar accuracy.Note that the thickness effects are relatively large for the normalized KII solution based on the computational results(Fig. 6). However, the effects of t/a cannot be accounted for by FI, FII, and FIII, and the coupling effects of W/a and L0/a cannotbe accounted for based on the interpolation method. The determination of FI, FII, and FIII is based on the computational resultsfor a small ratio of t/a = 0.2. Therefore, the deviation from 1.0 of the normalized KII solutions of Radaj et al. [20] and Zhang[21] may be partly due to the large ratios of t/a = 0.4 and 0.57, respectively. Another possible reason is that Radaj et al. [20]used the plate and brick elements to represent the sheets and weld nuggets, respectively. Zhang [21] treated the weld nuggetas rigid and modeled the sheet metals by plate elements. However, the finite element model investigated here use the brickelements for the whole specimen. Therefore, as t/a increases, the thickness effects increase and the computational solutionsof Radaj et al. [20] and Zhang [21] deviate from our computational results.

Another application of the accurate stress intensity factor solutions is to correlate the existing experimental data. Here,the experimental data for cross-tension specimens with different geometries in Mizui et al. [29], Lee and Kim [30], andHilditch et al. [31] are selected. Table 4 lists the average applied load ranges DF at different fatigue lives for the specimensof Mizui et al. [29] (marked by Mz), Lee and Kim [30] (Marked by LK1 and LK2), and Hilditch et al. [31] (marked by Hd). Notethat the LK1 and LK2 specimens represent those with large and small sheet thicknesses, respectively. As listed in Table 4, theload ranges of these specimens are quite different to each other. Then, the effects of specimen geometry are considered by FI,FII, and FIII. Table 5 lists FI, FII, and FIII values for t/a, W/a and L0/a of Mz, LK1, LK2, and Hd specimens. The geometry factors FI,FII, and FIII are obtained based on the aforementioned interpolation approach. Finally, Table 6 lists the equivalent stress inten-

sity factor ranges DKIeq (¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDKIÞ2 þ ðDKIIÞ2

q) at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b)

of Mz, LK1, LK2, and Hd specimens at different fatigue lives. Note that DKI and DKII are derived based on the load ranges DF in

Table 4The average load ranges DF at different fatigue lives for cross-tension specimens with different geometries in Mizui et al. [29] (marked by Mz), Lee and Kim [30](Marked by LK1 and LK2), and Hilditch et al. [31] (marked by Hd).

Fatigue life (cycles) 104 105 106 2.1 � 106

(DF)Mz (kN) N/A 0.98 0.64 0.57(DF)LK1 (kN) 2.12 1.0 0.6 N/A(DF)LK2 (kN) 1.54 0.64 0.29 N/A(DF)Hd (kN) 1.71 1.14 0.69 N/A

Table 5The FI, FII, and FIII values for different specimen geometries of Mz, LK1, LK2, and Hd specimens.

t/a W/a L0/a FI FII FIII

Mz 0.47 8.33 8.33 1.16 0.7 0.55LK1 0.47 8.47 8.47 1.15 0.69 0.54LK2 0.4 10.0 10.0 1.05 0.59 0.46Hd 0.33 5.81 5.81 1.36 1.39 1.12

Table 6

The equivalent stress intensity factor ranges DKIeq ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDKIÞ2 þ ðDKIIÞ2

q� �at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b) of Mz,

LK1, LK2, and Hd specimens at different fatigue lives.

Fatigue life (cycles) 104 105 106 2.1 � 106

(DKIeq)Mz (MPaffiffiffiffiffimp

) N/A 18.98 12.33 11.01

(DKIeq)LK1 (MPaffiffiffiffiffimp

) 41.45 19.52 10.95 N/A

(DKIeq)LK2 (MPaffiffiffiffiffimp

) 53.48 22.13 9.98 N/A

(DKIeq)Hd (MPaffiffiffiffiffimp

) 28.42 18.06 11.20 N/A

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Table 4, the geometric factors FI and FII in Table 5, and Eqs. (13) and (14). Also, the KIII solution is identically zero at the criticallocations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b) and therefore is neglected here. As shown in Tables 4 and 6,the experimental results for the specimens with different geometries in terms of DKIeq appear to be closer to each other ascompared to those in terms of DF. The accurate stress intensity factor solutions obtained from the three-dimensional finiteelement analyses appear to be a good fracture mechanics parameter to correlate the experimental fatigue data.

Note that the equivalent stress intensity factor ranges DKIeq at the fatigue life of 104 show a relatively large scatter com-pared to those at higher fatigue lives, as listed in Table 6. For spot welds under high-cycle fatigue conditions, the plastic zonesize at the crack tip in general is small compared with other geometric parameters of the specimen in lower load ranges butbecomes relatively large in higher load ranges. Based on the fracture mechanics, the effects of the crack tip plasticity on thestress intensity factors can be negligible for lower load ranges but appear to become significant for higher load ranges. There-fore, the fatigue behaviors of spot welds under higher load ranges may not correlate well with the stress intensity factors.Further investigation is needed to consider the effects of the crack tip plasticity on the stress intensity factors of spot welds.

5. Conclusions

Three-dimensional finite element analyses are conducted to investigate the stress intensity factor solutions for spot weldsat the critical locations in cross-tension specimens. The mode I stress intensity factor solution, the mode II stress intensityfactor solution at the critical locations (points CT01, CT02, CT03, and CT04 as shown in Fig. 1b) and the mode III stress inten-sity factor solution at the critical locations (points CT05, CT06, CT07, and CT08 as shown in Fig. 1b) for spot welds in cross-tension specimens are obtained by three-dimensional finite element computations. The solutions can be correlated withthose based on the analytical solutions of Zhang [3] with new geometric functions in terms of the normalized specimenwidth and normalized effective specimen length. The computational results confirm the functional dependence on the nug-get radius and sheet thickness of the stress intensity factor solutions of Zhang [3]. The computational results indicate that asthe normalized specimen width increases significantly, the geometric functions appear to be relatively constant. As the effec-tive specimen length increases significantly, the geometric functions also appear to be relatively constant; however, themode mixture changes significantly. The mode I stress intensity factor is the dominant factor only when the white rectan-gular areas of the specimen (Fig. 1a) are nearly square. Based on the analytical and computational results, the dimensions ofcross-tension specimens and the corresponding approximate stress intensity factor solutions are suggested. Note that thegeometric functions can be complex functions of the normalized sheet thickness, normalized specimen width, and normal-ized effectively specimen length. Therefore, consideration of the geometric functions for stress intensity factor solutions canbe useful to examine the existing fatigue data for spot welds in cross-tension specimens of various dimensions. The use ofthe stress intensity factor solutions to predict the fatigue lives of spot welds has been addressed in Newman and Dowling [8],Lin and Pan [32], Lin et al. [33–35], and Tran et al. [36,37] for fatigue life prediction of resistance spot welds and friction stirspot welds in various specimens.

Acknowledgments

The support of this work by the National Science Council under Grant No. 98-2221-E194-007-MY3 is greatly appreciated.The partial support of this work by China Steel Corporation (CSC) under Grant No. 98T1F-RE033 is also greatly appreciated.Helpful discussion with Dr. S. Zhang of Daimler AG is greatly appreciated.

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